Question: The sum of two angles is $81^\circ$. Angle 2 is $144^\circ$ smaller than $4$ times angle 1. What are the measures of the two angles in degrees?
Solution: Let $x$ equal the measure of angle 1 and $y$ equal the measure of angle 2. The system of equations is then: ${x+y = 81}$ ${y = 4x-144}$ Since we already have solved for $y$ in terms of $x$ , we can use substitution to solve for $x$ and $y$ Substitute ${4x-144}$ for $y$ in the first equation. ${x + }{(4x-144)}{= 81}$ Simplify and solve for $x$ $ x+4x - 144 = 81 $ $ 5x-144 = 81 $ $ 5x = 225 $ $ x = \dfrac{225}{5} $ ${x = 45}$ Now that you know ${x = 45}$ , plug it back into $ {y = 4x-144}$ to find $y$ ${y = 4}{(45)}{ - 144}$ $y = 180 - 144$ ${y = 36}$ You can also plug ${x = 45}$ into $ {x+y = 81}$ and get the same answer for $y$ ${(45)}{ + y = 81}$ ${y = 36}$ The measure of angle 1 is $45^\circ$ and the measure of angle 2 is $36^\circ$.